The Gunners were outclassed as they lost 2-0 to bitter rivals Tottenham, a result which confirmed they will finish below Spurs in the Premier League table for the first time since 1995.
Petr Cech was one of only a handful of players from the losing side to walk across to the away section at full-time, while the majority of the team headed straight for the tunnel.
Former Arsenal full-back Dixon tweeted: "No excuse for walking off the pitch without so much as a nod at the travelling fans. You know who you are!"
Arsenal look set to miss out on Champions League football for the first time in two decades, and manager Arsene Wenger is yet to confirm whether he will sign an extension to his current deal, which expires in the summer.
The Gunners have won just three of their last eight Premier League fixtures leading growing calls for Wenger to step aside at the end of the season.
No excuse for walking off the pitch without so much as a nod at the travelling fans. You know who you are!!— Lee Dixon (@LeeDixon2) April 30, 2017
And when it was put to Dixon that players' failure to acknowledge supporters was "the least of Arsenal's worries", he responded: "It's at the very root of our problems."